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Properties

Label 2-168-7.2-c1-0-2
Degree $2$
Conductor $168$
Sign $0.386 + 0.922i$
Analytic cond. $1.34148$
Root an. cond. $1.15822$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)5-s + (0.5 − 2.59i)7-s + (−0.499 + 0.866i)9-s + (−1.5 − 2.59i)11-s + 4·13-s − 0.999·15-s + (2 − 3.46i)19-s + (−2.5 + 0.866i)21-s + (−4 + 6.92i)23-s + (2 + 3.46i)25-s + 0.999·27-s − 3·29-s + (2.5 + 4.33i)31-s + (−1.5 + 2.59i)33-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.223 − 0.387i)5-s + (0.188 − 0.981i)7-s + (−0.166 + 0.288i)9-s + (−0.452 − 0.783i)11-s + 1.10·13-s − 0.258·15-s + (0.458 − 0.794i)19-s + (−0.545 + 0.188i)21-s + (−0.834 + 1.44i)23-s + (0.400 + 0.692i)25-s + 0.192·27-s − 0.557·29-s + (0.449 + 0.777i)31-s + (−0.261 + 0.452i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.902111 - 0.600068i\)
\(L(\frac12)\) \(\approx\) \(0.902111 - 0.600068i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 2.59i)T \)
good5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (-2.5 - 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + (5 - 8.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.5 - 4.33i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3 + 5.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7T + 83T^{2} \)
89 \( 1 + (-9 + 15.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 17T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.83942181260212742152509582866, −11.41031791247393288423130568905, −10.90354637271078615565653728121, −9.590736515814706101220057799992, −8.362824526134932600180029505650, −7.42004411163894806630952604738, −6.19750591618151433494307400364, −5.06532804769675224381579077430, −3.46525377428093816798725622350, −1.23009030174167561306806809656, 2.39487519051647458122973577984, 4.10350619475580917350151538029, 5.51090442417589854718756050162, 6.38018132140269793665373088371, 7.942831106571960374212948670689, 9.000423049924756827261221646976, 10.08768790591763689599556331058, 10.89161745293403720966204279901, 11.99571640242203150594276245619, 12.77700785914237666091944282905

Graph of the $Z$-function along the critical line